• Find A Slot Machine
• 777 Free Slots
• Real Free Slots
• A Slot Machine Is An Example Of A
• Gambling At A Slot Machine Is An Example Of
• Inside A Slot Machine

Slot machines use a variable ratio because. A) the gambler won't be able to tell when the next payoff is going to occur b) it increases the gambler's resistance to quitting c) the gambler will fear that.

The fundamentals of slot math. The take-away is to know why this math is so important in monitoring performance and compliance at your property. Slot machine math will assist in determining:. Whether a machine is performing as the vendor intended. The customers’ preferences for a machine on the floor. Introduction Slot machines are a very popular form of gambling in North America. For example, Ontario, Canada, has approximately 23,000 slot machines, which in the fiscal year 2002- 2003 generated approximately three billion dollars 'after prizes/winnings but before operating expenses' (Williams & Wood, 2004, p. The following is an example of a State Statute (Alabama) defining Slot Machine: Code of Ala. § 13A-12-20 (10) defines Slot Machine as “a gambling device that, as a result of the insertion of a coin or other object, operates, either completely automatically or with the aid of some physical act by the player, in such a manner that, depending.

Today, the mathematics of slot machines. The University of Houston mathematics department presents this program about the machines that make our civilization run, and the people whose ingenuity created them.

Mathematicians first got interested in randomness by studying games of chance. Ever since, the histories of mathematics and gambling have been intertwined. Clever gamblers use mathematics to look for the smallest advantages, and casinos use sophisticated mathematical tools to devise new ways of drawing in players.

Indeed, a patent granted to the Norwegian mathematician Inge Telnaes in 1984 transformed the gambling industry. Prior to Telnaes’ invention, slot machines were essentially mechanical devices. Besides being difficult to tune and maintain, mechanical slot machines suffered from an essential problem: Let’s look at a machine with three reels, each with 12 symbols, with one of those 12 symbols a cherry. The likelihood of getting three cherries, and winning the jackpot, is 1 in 1,728. If the casino wants to make money, the jackpot payout should be, say $1,700 on a $1 bet. That does not seem attractive by today’s standards. However, the only way to increase the payout is to decrease the chances of hitting a jackpot.

Adding another reel is a possibility. For instance adding a fourth reel in the previous example would get us to a jackpot of about $20,000. But people do not like machines with more reels — they intuitively, and rightfully, feel that extra reels diminish their chance of winning. Another possibility is to put more symbols on each reel. But the astronomical jackpots you see in casinos today would then require truly enormous machines.

Inge Telnaes proposed a simple solution: Let a random number generator — a computer chip — determine the combination of symbols that appear when the reels stop. In other words, use a chip to control where the reels stop on a spin, but create the illusion that the wheels stopped on their own. The number of possible outcomes on the slot machine does not change. However, by reprogramming the chip, the operator has full control over the likelihood of each of the different outcomes. For instance, the operator could make the three cherries appear only once in a million spins.

This was a brilliant insight: Suppose I pick a number between one and a million. Would you be willing to bet that you can guess that number? The answer is probably not. But let a computer chip pick such a number, put the chip in a machine with blinking lights and spinning reels, and many people will be more than willing to make the bet. It is simply because what people assume is happening in a slot machine is very different from what is actually happening.

The Magician oil painting by Hieronymus Boschfrom between 1475 and 1480

The history of gambling is also intertwined with that of a less reputable group — tricksters and swindlers. In the long run, the only sure way to make money by gambling is to create the illusion that your opponent can win, while keeping the odds firmly on your side. And that gives those who know math a very solid advantage.

I'm Krešimir Josić, at the University of Houston, where we're interested in the way inventive minds work.

(Theme music)
NOTE: In the example with three cherries, I assumed that one only wins in the case the spin results in three cherries, and there is no other winning combination. In actuality, there are typically many winning combination, and as a result, the jackpot would have to be even smaller.

The following story in Wired Magazine shows the drawbacks of the new generation of slot machines — they are easier to hack and to counterfit than their mechanical counterpart http://www.wired.com/magazine/2011/07/ff_scammingslots/.

Here is a more exhaustive discussion of the history of slot machines, and the random number generators within them http://catlin.casinocitytimes.com/article/non-random-randomness-part-1-1243. You may want to scroll towards the end of the article to read about how flaws in the design of gambling machine resulted in somebody picking 19 out of 20 winning numbers in a game of KENO — and doing so 3 times in a row. That person walked away with $620,000, but only after some controversy.

Both images are from Wikipedia. The slot machine image was taken by Jeff Kubina.

For more mathematics in everyday life, visitkjosic.wordpress.com.

This episode was first aired on September 7th, 2011

Find A Slot Machine

• Appendices
• Slots Analysis
• Miscellaneous

Introduction

Click on this image to play.

I designed the Atkins Diet slot machine in 2008 for a speaking engagement at a conference on slot machines in Sydney, Australia. It was used as an example of how video slots, sometimes known as pokies, are designed. I'll start with presenting the pay table and reel strips, and then I'll show the probability and return of the various ways to win.

Pay Table

The next table shows the Atkins Diet pay table. All wins must be left aligned. The Atkins symbol is wild and can substitute for any symbol except the scale.

Atkins Diet Pay Table

Symbol	5 in a row	4 in a row	3 in a row	2 in a row
Atkins	5000	500	50	5
Steak	1000	200	40	3
Ham	500	150	30	2
Buffalo Wings	300	100	25	2
Sausage	200	75	20	0
Eggs	200	75	20	0
Butter	100	50	15	0
Cheese	100	50	15	0
Bacon	50	25	10	0
Mayonnaise	50	25	10	0

Reel Strips

The following are the reel strips for Atkins Diet.

Atkins Diet Reel Strips

Stop	Reel 1	Reel 2	Reel 3	Reel 4	Reel 5
1	Scale	Mayonnaise	Ham	Ham	Bacon
2	Mayonnaise	Buffalo Wings	Butter	Cheese	Scale
3	Ham	Steak	Eggs	Atkins	Steak
4	Sausage	Sausage	Scale	Scale	Ham
5	Bacon	Cheese	Cheese	Butter	Cheese
6	Eggs	Mayonnaise	Mayonnaise	Bacon	Sausage
7	Cheese	Ham	Butter	Cheese	Butter
8	Mayonnaise	Butter	Ham	Sausage	Bacon
9	Sausage	Bacon	Sausage	Steak	Buffalo Wings
10	Butter	Steak	Bacon	Eggs	Cheese
11	Buffalo Wings	Sausage	Steak	Bacon	Sausage
12	Bacon	Mayonnaise	Buffalo Wings	Mayonnaise	Ham
13	Eggs	Ham	Butter	Sausage	Butter
14	Mayonnaise	Atkins	Mayonnaise	Cheese	Steak
15	Steak	Butter	Cheese	Butter	Mayonnaise
16	Buffalo Wings	Eggs	Sausage	Ham	Eggs
17	Butter	Cheese	Eggs	Mayonnaise	Sausage
18	Cheese	Bacon	Bacon	Bacon	Ham
19	Eggs	Sausage	Mayonnaise	Buffalo Wings	Atkins
20	Atkins	Buffalo Wings	Buffalo Wings	Sausage	Butter
21	Bacon	Scale	Ham	Cheese	Buffalo Wings
22	Mayonnaise	Mayonnaise	Sausage	Eggs	Mayonnaise
23	Ham	Butter	Bacon	Butter	Eggs
24	Cheese	Cheese	Cheese	Buffalo Wings	Ham
25	Eggs	Bacon	Eggs	Bacon	Bacon
26	Scale	Eggs	Atkins	Mayonnaise	Butter
27	Butter	Buffalo Wings	Buffalo Wings	Eggs	Steak
28	Bacon	Mayonnaise	Bacon	Ham	Mayonnaise
29	Sausage	Steak	Butter	Sausage	Sausage
30	Buffalo Wings	Ham	Cheese	Steak	Eggs
31	Steak	Cheese	Mayonnaise	Mayonnaise	Cheese
32	Butter	Bacon	Steak	Bacon	Buffalo Wings

Example Spin

Let's look at the following example spin.

The way multi-line video slots work is to pick one random number for each reel and map it onto the corresponding stop on the reel strip. Let's assume that the random number chosen designates where the reel will stop on the middle row. The symbols above and below will be those directly above and below on the reel strips.

Random numbers of 27, 14, 3, 31 and 27, for reels 1 to 5 respectively, would have resulted in the example above. The following table shows the position on the reel strips where all 15 visible symbols fall on the screen.

Atkins Diet Reel Strips

Stop	Reel 1	Reel 2	Reel 3	Reel 4	Reel 5
Top	26	13	2	30	26
Middle	27	14	3	31	27
Bottom	28	15	4	32	28

It should also be noted that the reel strips wrap around. So, if the number chosen for the middle position is 1, then the top position would be 32. Likewise, if the middle position is 32, then the bottom position would be 1.

Symbol Distribution

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The first step to calculating the return of a video slot is to count how often each symbol appears on each reel. The following table answers that question.

Atkins Diet Symbol Distribution

Symbol	Reel 1	Reel 2	Reel 3	Reel 4	Reel 5
Atkins	1	1	1	1	1
Steak	2	3	2	2	3
Ham	2	3	3	3	4
Buffalo Wings	3	3	3	2	3
Sausage	3	3	3	4	4
Eggs	4	2	3	3	3
Butter	4	3	4	3	4
Cheese	3	4	4	4	3
Bacon	4	4	4	5	3
Mayonnaise	4	5	4	4	3
Scale	2	1	1	1	1
Total	32	32	32	32	32

Line Pay Math

There are 32 × 32 × 32 × 32 × 32 = 33,554,432 possible outcomes in Atkins Diet. The following table shows how many of these combinations result in each possible win. These figures are the result of a program with five nested loops that tallied the total for each win for each possible combination.

Atkins Diet Line Pay Combinations

Symbol	5 in a row	4 in a row	3 in a row	2 in a row	Total
Atkins	1	28	513	1,024	1,566
Steak	431	2,996	32,480	326,656	362,563
Ham	955	5,157	42,112	315,392	363,616
Buffalo Wings	764	5,348	58,464	430,080	494,656
Sausage	1,595	8,613	54,432	-	64,640
Eggs	956	6,692	52,864	-	60,512
Butter	1,995	10,692	88,704	-	101,391
Cheese	1,996	13,860	85,536	-	101,392
Bacon	2,976	20,832	103,168	-	126,976
Mayonnaise	2,980	20,860	128,736	-	152,576
Total	14,649	95,078	647,009	1,073,152	1,829,888

The next table shows the probability of each possible win. This is the number of winning combinations for that win, shown in the table above, divided by the total number of combinations of 33,554,432. The total in the lower right cell reflects a hit frequency per line (probability of winning anything) of 5.45%.

Atkins Line Pay Probabilities

Symbol	5 in a row	4 in a row	3 in a row	2 in a row	Total
Atkins	0.00000003	0.00000083	0.00001529	0.00003052	0.00004667
Steak	0.00001284	0.00008929	0.00096798	0.00973511	0.01080522
Ham	0.00002846	0.00015369	0.00125504	0.00939941	0.01083660
Buffalo Wings	0.00002277	0.00015938	0.00174236	0.01281738	0.01474190
Sausage	0.00004753	0.00025669	0.00162220	-	0.00192642
Eggs	0.00002849	0.00019944	0.00157547	-	0.00180340
Butter	0.00005946	0.00031865	0.00264359	-	0.00302169
Cheese	0.00005949	0.00041306	0.00254917	-	0.00302172
Bacon	0.00008869	0.00062084	0.00307465	-	0.00378418
Mayonnaise	0.00008881	0.00062168	0.00383663	-	0.00454712
Total	0.00043657	0.00283355	0.01928237	0.03198242	0.05453491

The next table shows the contribution to the return for each possible win. Each cell is the product of the probability, from the table above, and the pay table. The right cell shows a total return from line pays of 63.46%.

Atkins Diet Expected Returns

Symbol	5 in a row	4 in a row	3 in a row	2 in a row	Total
Atkins (wild)	0.000149	0.000417	0.000764	0.000153	0.001483
Steak	0.012845	0.017858	0.038719	0.029205	0.098627
Ham	0.014231	0.023054	0.037651	0.018799	0.093734
Buffalo Wings	0.006831	0.015938	0.043559	0.025635	0.091963
Sausage	0.009507	0.019252	0.032444	-	0.061202
Eggs	0.005698	0.014958	0.031509	-	0.052165
Butter	0.005946	0.015932	0.039654	-	0.061532
Cheese	0.005949	0.020653	0.038238	-	0.064839
Bacon	0.004435	0.015521	0.030746	-	0.050702
Mayonnaise	0.004441	0.015542	0.038366	-	0.058349
Total	0.070029	0.159124	0.331651	0.073792	0.634597

Scatter Pay

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The scale serves as a scatter pay symbol. The player gets credit for a scatter pay if it appears anywhere on the screen. Unlike line pays, scatters don't need to be left aligned or on an active pay line. To be fair, scatter pay wins are based on the total amount bet. The following table shows the number of scatters per reel, total symbols per reel, and the probability of a scatter on that reel, which is 3×(column 2)/(column 3). The reason for multiplying by 3 is there are three visible positions on each reel.

Atkins Diet Scatters per Reel

Reel	Scatters	Total Symbols on Reel	Scatter Probability
1	2	32	0.18750
2	1	32	0.09375
3	1	32	0.09375
4	1	32	0.09375
5	1	32	0.09375

A Slot Machine Is An Example Of A

Gambling At A Slot Machine Is An Example Of

The next table shows the probability and contribution to the return for 0 to 5 total scatter symbols. The lower right cell shows that the scatter pay feature returns 6.98% to the return of the game.

Atkins Diet Scatter Pay Probabilities and Returns

Scatters	Pays	Combinations	Probability	Return
5	100	486	0.000014	0.001448
4	25	20,898	0.000623	0.015570
3	5	353,916	0.010548	0.052738
Total	33,554,432	0.011185	0.069756

Inside A Slot Machine

Bonus

The player earns the bonus if he gets at least three scattered scales. The sum of the probability of 3 to 5 scales in the table above is 0.011185. During the bonus, the player will get ten free spins, and all wins are tripled.

In this game, free spins use the same reel strips as the base game. We have already shown that the expected line pay return is 0.634597 and the expected scatter return is 0.069756. So, if the win is tripled, then the expected win per spin is 3 × (0.634597 + 0.069756) = 2.113058.

Next, there is a rule that free spins can be won in free spins, infinitely. The expected number of total spins in a bonus is 10/(1-10×0.011185) = 11.259335.

So, with 11.259335 free spins, and an average win per spin of 2.113058, the expected final win per bonus is 11.259335 × 2.113058 = 23.791632.

The probability of winning the bonus in the first place is 0.011185, so the expected return per initial spin of the bonus feature is 0.011185 × 23.791632 = 0.266105.

Summary

The following table summarizes the three ways to win in the game. The bottom right cell shows an expected return of 97.046%. In other words, for every dollar the player bets, he can expect to get back 97¢.

Atkins Diet Summary

Item	Return
Line Pays	63.460%
Scatter	6.976%
Bonus	26.610%
Total	97.046%

More Deconstructions

I have deconstructed lots of slot machines. Here is a list of them:

• Deconstructing Jackpot Party analysis of the video slot machine.
  
• Deconstructing Lion's Share analysis of the classic MGM progressive game.
  
• Deconstructing Cleopatra analysis of the popular IGT game.
  
• Deconstructing Lionfish analysis of the slot game found on many Game Maker machines.
  
• Deconstructing Megabucks.
• Deconstructing the Atkins Diet slot machine.
• Deconstructing Lucky Larry's Lobstermania.
• Deconstructing Hexbreaker.
• Deconstructing Blazing Sevens.
• Deconstructing Hot Roll.

Internal Links

• Play Atkins Diet. Demo you can play for fun.
• Deconstructing the Atkins Diet Slot Machine. Earlier PDF version of this page.
• Slot machines main page.
• Appendix 4 shows how the return is calculated for my Wizard's Fruit Slot Machine.
  
• Appendix 5 analysis of the 21 Bell Slot Machine.
  
• Appendix 6 Analysis of Red, White, & Blue Slot Machine.
  
• Lock and Roll analysis of the skill-based slot machine found in North Carolina.
  

Written by: Michael Shackleford